Lie algebra and invariant tensor technology for g2

TL;DR

This paper develops the tensor and invariant tensor technology for the Lie algebra g2, providing identities and relations crucial for understanding its structure and representations.

Contribution

It introduces a comprehensive framework for g2's invariant tensors, extending the analogy with su(n) and detailing identities involving Casimir operators.

Findings

Complete set of identities for g2 invariant tensors

Relations involving quadratic and sextic Casimir operators

Characterization of the non-primitivity of certain Casimir operators

Abstract

Proceeding in analogy with su(n) work on lambda matrices and f- and d-tensors, this paper develops the technology of the Lie algebra g2, its seven dimensional defining representation gamma and the full set of invariant tensors that arise in relation thereto. A comprehensive listing of identities involving these tensors is given. This includes identities that depend on use of characteristic equations, especially for gamma, and a good body of results involving the quadratic, sextic and (the non-primitivity of) other Casimir operators of g2.